Topics in Topology ("Topologie III"), Sommersemester 2025

lecture notes The actual notes for this course begin with Chapter 56, on page 493. The notes were last updated on 27.06.2025 and now contain everything up to the lecture of 26.06.2025 (stable homotopy groups and spectra). You may want to press the reload button to make sure you are seeing the current version. What appears before Chapter 56 is a complete set of notes that I wrote for Topology I-II in previous semesters. This is convenient to have there so that I can occasionally refer to it in the current course, but that does not mean that you must already know most of what is contained in the earlier parts of the notes. (See Prerequisites below for more on that.) Problem Sets Problem sets are generally posted here by the end of each week and are to be discussed in the Übung the following week. Exercises in the lecture notes that are marked with an asterisk should be considered essential; they typically concern proofs of results that will be used later in the course. Problem Set 1 (to be discussed 24.04.2025): All exercises in Chapters 56 and 57 of the lecture notes. Problem Set 2 (to be discussed 15.05.2025): All exercises in Chapters 58, 59, 60 and 61 of the lecture notes. Note: This set is a special case, because we had two Thursdays in a row that were holidays. Since it would be impossible to cover the problems from four lectures in one problem session, written solutions to select problems have now been posted on the moodle. Problem Set 3 (to be discussed 22.05.2025): All exercises in Chapters 62 and 63 of the lecture notes. Problem Set 4 (to be discussed 5.06.2025): All exercises in Chapters 64, 65 and 66 of the lecture notes Problem Set 5 (to be discussed 12.06.2025): All exercises in Chapters 67 and 68 of the lecture notes Problem Set 6 (to be discussed 19.06.2025): All exercises in Chapters 69 and 70 of the lecture notes Problem Set 7 (to be discussed 26.06.2025): All exercises in Chapters 71 and 72 of the lecture notes Note: Exercise 71.3 used to appear in Chapter 69 and was already discussed in the session on 19.06. I've displaced it now because I'd forgotten that its solution uses the simply connected case of the relative Hurewicz theorem. Problem Set 8 (to be discussed 3.07.2025): All exercises in Chapters 73 and 74 of the lecture notes For general information about the course, scroll down a bit... Announcements 25.04.2025: A major adjustment has been made to the schedule for the Übung, and also minor adjustments for the lectures; see the red text below under "Time and place". 13.04.2025: The problem class (Übung) will not meet in the first week of the semester.

image by Christian Lawson-Perfect from cp's mathem-o-blog

General information

Instructors:

• Prof. Chris Wendl (lectures); for contact information and office hours see my homepage
• Naageswaran Manikandan (problem class)

Moodle:
https://moodle.hu-berlin.de/course/view.php?id=133147
The enrollment key is: obstruction
Important: You must join the moodle for the course in order to receive occasional time-sensitive announcements, e.g. if a lecture has been cancelled or rescheduled. HU students can access moodle using their HU username and password. Non-HU users can access it by following this link and then clicking on "Create new account". You will need to enter the enrollment key printed above.

Time and place:
Lectures on Wednesdays and Thursdays 11:00-12:30 in room 1.012 (RUD25)
Problem Class (Übung) Thursdays 13:30-15:00 in room 1.012 (RUD25), starting in the second week of the semester
Note: Some minor adjustments have been made to the schedule since the start of the semester, and are indicated above in red. Note in particular that both lectures start on the hour, not 15 minutes after.

Language:
The course will be taught in English.

Prerequisites:
The main prerequisites are a solid foundation in point-set topology, the fundamental group and covering spaces, singular and cellular (co-)homology (including computations based on the axioms and some homological algebra, e.g. the universal coefficient theorems), and some willingness to put up with the language of categories, functors and universal properties. If you have taken my Topologie II course before, then you definitely have the essential prerequisites, but you probably also have them if you learned about homology and cohomology elsewhere. Some knowledge of smooth manifolds will occasionally be useful, but if you do not have this, you will just need to be willing to accept a small set of facts about tangent spaces, tubular neighborhoods and transversality as black boxes.

Contents:
This is a course on intermediate-level algebraic topology, and is conceived in part as a sequel to last semester's Topology 2 course. We aim to prove some useful results from elementary homotopy theory (some of which were mentioned briefly last semester but were not proved), and introduce the essentials of obstruction theory, classifying spaces, characteristic classes, and bordism theory. These are all topics that have wide-ranging applications to other areas of mathematics, especially to problems in differential geometry and topology, such as the existence and classification of exotic smooth structures on manifolds. We will not attempt any deep exploration of modern homotopy theory, as that would far exceed my expertise.

Here is a more detailed plan of topics, though I cannot promise that all of them will be covered.

• Notions from category theory: limits and colimits, (co-)products, fiber products / pullbacks and pushouts, (co-)equalizers
• Point-set topological subtleties: evaluation maps, adjoint functors, quotient maps, the compactly-generated category
• Elementary homotopy theory: mapping cylinders and cones, reduced suspensions, loop spaces and adjunction, fibrations and cofibrations, homotopy (co)fibers, transport functors, the Puppe sequences, group and cogroup objects
• Higher homotopy groups: definitions and group structure, exact sequences, Serre fibrations, homotopy excision theorem, Freudenthal suspension theorem
• Homotopy theory of CW-complexes: weak homotopy equivalences and Whitehead's theorem, n-connectedness, cellular and CW-approximation, Eilenberg-MacLane spaces, homotopical characterization of cellular cohomology, Hurewicz theorem
• Generalized (co)homology theories: (co)homology for pointed spaces, stable homotopy groups, spectra
• Topology of fiber bundles: vector/principal/fiber bundles and structure groups, induced bundles and homotopy invariance, universal bundles and classifying spaces, reduction of structure groups, stable equivalence and K-groups
• Obstruction theory: singular and cellular (co-)homology with local coefficients, obstruction cocycles for lifting/extension problems
• Characteristic classes: Chern, Stiefel-Whitney, Pontryagin and Euler classes, the splitting principle, basic computations
• Bordism theory: definitions as generalized homology, stable Pontryagin-Thom theorem, tangential structures, computation of the rational oriented bordism ring, Hirzebruch signature theorem

Literature:
The top of this page contains a link to detailed lecture notes for this course which will be updated routinely as the course progresses. The bulk of what we plan to cover is in any case contained in the union of the following three books:

• Tammo tom Dieck: Algebraic Topology (EMS, 2010)
• James F. Davis and Paul Kirk: Lecture Notes in Algebraic Topology (AMS, 2001)
• J. P. May: A Concise Course in Algebraic Topology (U. Chicago Press, 1999); also downloadable from Peter May's homepage

• Tyrone Cutler's course on Homotopy Theory (from Bielefeld, 2021)
• George W. Whitehead: Elements of Homotopy Theory (Springer GTM, 1978)
• Norman Steenrod: The Topology of Fibre Bundles (Princeton, 1951). A bit antiquated, but a timeless classic... this is the original source for obstruction theory.
• Daniel S. Freed: Bordism, Old and New (lecture notes, 2012). An excellent source both for bordism theory and several other topics, making somewhat more liberal use of smoothness (i.e. methods from differential topology) than the typical book on "algebraic" topology does.

• Allen Hatcher: Algebraic Topology (Cambridge, 2002); you'll never be free of it. This recommendation comes with the caveat that if you want to learn about products in singular (co-)homology but get all the signs right, then you should look at a different book.
• Raoul Bott and Loring W. Tu: Differential Forms in Algebraic Topology (Springer GTM, 1982). If days were 36 hours long instead of 24, I would teach a whole parallel course following this book.
• Edwin H. Spanier: Algebraic Topology (Springer, 1966). Every time I think I am finished learning new things from this book, I turn out to be wrong.

Homework:
Weekly problem sets will be posted near the top of this page, and some subset of those problems will be discussed in the problem class (Übung) each week. Homework will not be collected or graded. The problem class may sometimes also be used to fill in gaps on details that did not fit into the regular lectures.

Grades:
Since this is an advanced course, I have a fairly relaxed attitude about grades. If you come to the course with adequate prerequisites and stay with it for the whole semester, you can come to my office at the end for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). The format is as follows: you pick one particular coherent topic from the course to focus on, typically the contents of four to six lectures, and we will talk about that. If you demonstrate that you learned something interesting from the course, you'll get a good grade.